Functions and graphs - Grade 10 [CAPS] - CNX

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Functions and graphs - Grade 10[CAPS]*

Free High School Science Texts Project

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Free High School Science Texts Project

Heather Williams

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1 Introduction to Functions and Graphs

Functions are mathematical building blocks for designing machines, predicting natural disasters, curingdiseases, understanding world economies and for keeping aeroplanes in the air. Functions can take inputfrom many variables, but always give the same answer, unique to that function. It is the fact that you alwaysget the same answer from a set of inputs that makes functions special.

A major advantage of functions is that they allow us to visualise equations in terms of a graph. A graphis an accurate drawing of a function and is much easier to read than lists of numbers. In this chapter we willlearn how to understand and create real valued functions, how to read graphs and how to draw them.

In addition to their use in the problems facing humanity, functions also appear on a day-to-day level, sothey are worth learning about. A function is always dependent on one or more variables, like time, distanceor a more abstract quantity.

2 Functions and Graphs in the Real-World

Some typical examples of functions you may already have met include:-

• how much money you have, as a function of time. You never have more than one amount of moneyat any time because you can always add everything to give one number. By understanding how yourmoney changes over time, you can plan to spend your money sensibly. Businesses �nd it very useful toplot the graph of their money over time so that they can see when they are spending too much. Suchobservations are not always obvious from looking at the numbers alone.

• the temperature is a very complicated function because it has so many inputs, including; the time ofday, the season, the amount of clouds in the sky, the strength of the wind, where you are and many

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more. But the important thing is that there is only one temperature when you measure it in a speci�cplace. By understanding how the temperature is e�ected by these things, you can plan for the day.

• where you are is a function of time, because you cannot be in two places at once! If you were to plot

the graphs of where two people are as a function of time, if the lines cross it means that the two peoplemeet each other at that time. This idea is used in logistics, an area of mathematics that tries to planwhere people and items are for businesses.

• your weight is a function of how much you eat and how much exercise you do, but everybody has adi�erent function so that is why people are all di�erent sizes.

3 Recap

The following should be familiar.

3.1 Variables and Constants

In Review of past work, we were introduced to variables and constants. To recap, a variable can take anyvalue in some set of numbers, so long as the equation is consistent. Most often, a variable will be written asa letter.

A constant has a �xed value. The number 1 is a constant. Sometimes letters are used to representconstants, as they are easier to work with.

3.1.1 Investigation : Variables and Constants

In the following expressions, identify the variables and the constants:

1. 2x2 = 12. 3x+ 4y = 73. y = −5

x4. y = 7x − 2

3.2 Relations and Functions

In earlier grades, you saw that variables can be related to each other. For example, Alan is two years olderthan Nathan. Therefore the relationship between the ages of Alan and Nathan can be written as A = N +2,where A is Alan's age and N is Nathan's age.

In general, a relation is an equation which relates two variables. For example, y = 5x and y2 + x2 = 5are relations. In both examples x and y are variables and 5 is a constant, but for a given value of x the valueof y will be very di�erent in each relation.

Besides writing relations as equations, they can also be represented as words, tables and graphs. Insteadof writing y = 5x, we could also say �y is always �ve times as big as x�. We could also give the followingtable:

x y = 5x

2 10

6 30

8 40

13 65

15 75

Table 1



3.2.1 Investigation : Relations and Functions

Complete the following table for the given functions:

x y = x y = 2x y = x+ 2

1

2

3

50

100

Table 2

3.3 The Cartesian Plane

When working with real valued functions, our major tool is drawing graphs. In the �rst place, if we havetwo real variables, x and y, then we can assign values to them simultaneously. That is, we can say �let x be 5and y be 3�. Just as we write �let x = 5� for �let x be 5�, we have the shorthand notation �let (x, y) = (5, 3)�for �let x be 5 and y be 3�. We usually think of the real numbers as an in�nitely long line, and picking anumber as putting a dot on that line. If we want to pick two numbers at the same time, we can do somethingsimilar, but now we must use two dimensions. What we do is use two lines, one for x and one for y, androtate the one for y, as in Figure 1. We call this the Cartesian plane.



Figure 1: The Cartesian plane is made up of an x−axis (horizontal) and a y−axis (vertical).

3.4 Drawing Graphs

In order to draw the graph of a function, we need to calculate a few points. Then we plot the points on theCartesian Plane and join the points with a smooth line.

Assume that we were investigating the properties of the function f (x) = 2x. We could then consider allthe points (x; y) such that y = f (x), i.e. y = 2x. For example, (1; 2) , (2, 5; 5) , and (3; 6) would all be suchpoints, whereas (3; 5) would not since 5 6= 2× 3. If we put a dot at each of those points, and then at every



similar one for all possible values of x, we would obtain the graph shown in Figure 2

Figure 2: Graph of f (x) = 2x

The form of this graph is very pleasing � it is a simple straight line through the middle of the plane. Thetechnique of �plotting�, which we have followed here, is the key element in understanding functions.

3.4.1 Investigation : Drawing Graphs and the Cartesian Plane

Plot the following points and draw a smooth line through them. (-6; -8),(-2; 0), (2; 8), (6; 16)

3.5 Notation used for Functions

Thus far you would have seen that we can use y = 2x to represent a function. This notation however getsconfusing when you are working with more than one function. A more general form of writing a function isto write the function as f (x), where f is the function name and x is the independent variable. For example,f (x) = 2x and g (t) = 2t+ 1 are two functions.

Both notations will be used in this book.

Exercise 1: Function notation (Solution on p. 31.)

If f (n) = n2 − 6n+ 9, �nd f (k − 1) in terms of k.

Exercise 2: Function notation (Solution on p. 31.)

If f (x) = x2 − 4, calculate b if f (b) = 45.

3.5.1 Recap

1. Guess the function in the form y = ... that has the values listed in the table.

x 1 2 3 40 50 600 700 800 900 1000

y 1 2 3 40 50 600 700 800 900 1000

Table 3

Click here for the solution1


x 1 2 3 40 50 600 700 800 900 1000

y 2 4 6 80 100 1200 1400 1600 1800 2000

Table 4

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x 1 2 3 40 50 600 700 800 900 1000

y 10 20 30 400 500 6000 7000 8000 9000 10000

Table 5


4. On a Cartesian plane, plot the following points: (1;2), (2;4), (3;6), (4;8), (5;10). Join the points. Doyou get a straight line?Click here for the solution4

5. If f (x) = x+ x2, write out:

a. f (t)b. f (a)c. f (1)d. f (3)


6. If g (x) = x and f (x) = 2x, write out:

a. f (t) + g (t)b. f (a)− g (a)c. f (1) + g (2)d. f (3) + g (s)


7. A car drives by you on a straight highway. The car is travelling 10 m every second. Completethe table below by �lling in how far the car has travelled away from you after 5, 10 and 20 sec-onds.

Time (s) 0 1 2 5 10 20

Distance (m) 0 10 20

Table 6

Use the values in the table and draw a graph of distance on the y-axis and time on the x-axis. Clickhere for the solution7

4 Characteristics of Functions - All Grades

There are many characteristics of graphs that help describe the graph of any function. These properties willbe described in this chapter and are:

1. dependent and independent variables2. domain and range

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3. intercepts with axes4. turning points5. asymptotes6. lines of symmetry7. intervals on which the function increases/decreases8. continuous nature of the function

Some of these words may be unfamiliar to you, but each will be clearly described. Examples of theseproperties are shown in Figure 3.

Figure 3: (a) Example graphs showing the characteristics of a function. (b) Example graph showingasymptotes of a function. The asymptotes are shown as dashed lines.

4.1 Dependent and Independent Variables

Thus far, all the graphs you have drawn have needed two values, an x-value and a y-value. The y-value isusually determined from some relation based on a given or chosen x-value. These values are given specialnames in mathematics. The given or chosen x-value is known as the independent variable, because its valuecan be chosen freely. The calculated y-value is known as the dependent variable, because its value dependson the chosen x-value.

4.2 Domain and Range

The domain of a relation is the set of all the x values for which there exists at least one y value accordingto that relation. The range is the set of all the y values, which can be obtained using at least one x value.

If the relation is of height to people, then the domain is all living people, while the range would be about0,1 to 3 metres � no living person can have a height of 0m, and while strictly it's not impossible to be tallerthan 3 metres, no one alive is. An important aspect of this range is that it does not contain all the numbersbetween 0,1 and 3, but at most six billion of them (as many as there are people).

As another example, suppose x and y are real valued variables, and we have the relation y = 2x. Thenfor any value of x, there is a value of y, so the domain of this relation is the whole set of real numbers.However, we know that no matter what value of x we choose, 2x can never be less than or equal to 0. Hencethe range of this function is all the real numbers strictly greater than zero.

These are two ways of writing the domain and range of a function, set notation and interval notation.Both notations are used in mathematics, so you should be familiar with each.

4.2.1 Set Notation

A set of certain x values has the following form:

x : conditions, more conditions (3)

We read this notation as �the set of all x values where all the conditions are satis�ed�. For example, the setof all positive real numbers can be written as {x : x ∈ R, x > 0} which reads as �the set of all x values wherex is a real number and is greater than zero�.



4.2.2 Interval Notation

Here we write an interval in the form 'lower bracket, lower number, comma, upper number, upper bracket'.We can use two types of brackets, square ones [; ] or round ones (; ). A square bracket means including thenumber at the end of the interval whereas a round bracket means excluding the number at the end of theinterval. It is important to note that this notation can only be used for all real numbers in an interval. Itcannot be used to describe integers in an interval or rational numbers in an interval.

So if x is a real number greater than 2 and less than or equal to 8, then x is any number in the interval

(2; 8] (3)

It is obvious that 2 is the lower number and 8 the upper number. The round bracket means 'excluding 2',since x is greater than 2, and the square bracket means 'including 8' as x is less than or equal to 8.

4.3 Intercepts with the Axes

The intercept is the point at which a graph intersects an axis. The x-intercepts are the points at which thegraph cuts the x-axis and the y-intercepts are the points at which the graph cuts the y-axis.

In Figure 3(a), the A is the y-intercept and B, C and F are x-intercepts.You will usually need to calculate the intercepts. The two most important things to remember is that

at the x-intercept, y = 0 and at the y-intercept, x = 0.For example, calculate the intercepts of y = 3x+5. For the y-intercept, x = 0. Therefore the y-intercept

is yint = 3 (0) + 5 = 5. For the x-intercept, y = 0. Therefore the x-intercept is found from 0 = 3xint + 5,giving xint = − 5

3 .

4.4 Turning Points

Turning points only occur for graphs of functions whose highest power is greater than 1. For example, graphsof the following functions will have turning points.

f (x) = 2x2 − 2

g (x) = x3 − 2x2 + x− 2

h (x) = 23x

4 − 2

(3)

There are two types of turning points: a minimal turning point and a maximal turning point. A minimalturning point is a point on the graph where the graph stops decreasing in value and starts increasing in valueand a maximal turning point is a point on the graph where the graph stops increasing in value and startsdecreasing. These are shown in Figure 4.

Figure 4: (a) Maximal turning point. (b) Minimal turning point.

In Figure 3(a), E is a maximal turning point and D is a minimal turning point.



4.5 Asymptotes

An asymptote is a straight or curved line, which the graph of a function will approach, but never touch.In Figure 3(b), the y-axis and line h are both asymptotes as the graph approaches both these lines, but

never touches them.

4.6 Lines of Symmetry

Graphs look the same on either side of lines of symmetry. These lines may include the x- and y- axes. Forexample, in Figure 5 is symmetric about the y-axis. This is described as the axis of symmetry. Not everygraph will have a line of symmetry.

Figure 5: Demonstration of axis of symmetry. The y-axis is an axis of symmetry, because the graphlooks the same on both sides of the y-axis.

4.7 Intervals on which the Function Increases/Decreases

In the discussion of turning points, we saw that the graph of a function can start or stop increasing ordecreasing at a turning point. If the graph in Figure 3(a) is examined, we �nd that the values of thegraph increase and decrease over di�erent intervals. We see that the graph increases (i.e. that the y-valuesincrease) from -∞ to point E, then it decreases (i.e. the y-values decrease) from point E to point D and thenit increases from point D to +∞.

4.8 Discrete or Continuous Nature of the Graph

A graph is said to be continuous if there are no breaks in the graph. For example, the graph in Figure 3(a)can be described as a continuous graph, while the graph in Figure 3(b) has a break around the asymptoteswhich means that it is not continuous. In Figure 3(b), it is clear that the graph does have a break in itaround the asymptote.

4.8.1 Domain and Range

1. The domain of the function f (x) = 2x+5 is -3; -3; -3; 0. Determine the range of f . Click here for thesolution8

2. If g (x) = −x2 + 5 and x is between - 3 and 3, determine:

a. the domain of g (x)b. the range of g (x)


3. On the following graph label:

a. the x-intercept(s)b. the y-intercept(s)

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c. regions where the graph is increasingd. regions where the graph is decreasing

Figure 6


4. On the following graph label:

a. the x-intercept(s)b. the y-intercept(s)c. regions where the graph is increasingd. regions where the graph is decreasing

Figure 7


5 Graphs of Functions

5.1 Functions of the form y = ax+ q

Functions with a general form of y = ax+ q are called straight line functions. In the equation, y = ax+ q, aand q are constants and have di�erent e�ects on the graph of the function. The general shape of the graphof functions of this form is shown in Figure 8 for the function f (x) = 2x+ 3.

Figure 8: Graph of f (x) = 2x+ 3

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5.1.1 Investigation : Functions of the Form y = ax+ q

1. On the same set of axes, plot the following graphs:

a. a (x) = x− 2b. b (x) = x− 1c. c (x) = xd. d (x) = x+ 1e. e (x) = x+ 2

Use your results to deduce the e�ect of di�erent values of q on the resulting graph.2. On the same set of axes, plot the following graphs:

a. f (x) = −2 · xb. g (x) = −1 · xc. h (x) = 0 · xd. j (x) = 1 · xe. k (x) = 2 · x

Use your results to deduce the e�ect of di�erent values of a on the resulting graph.

You may have noticed that the value of a a�ects the slope of the graph. As a increases, the slope of the graphincreases. If a > 0 then the graph increases from left to right (slopes upwards). If a < 0 then the graphincreases from right to left (slopes downwards). For this reason, a is referred to as the slope or gradient ofa straight-line function.

You should have also found that the value of q a�ects where the graph passes through the y-axis. Forthis reason, q is known as the y-intercept.

These di�erent properties are summarised in Table 7.

a > 0 a < 0

q > 0

Figure 9 Figure 10

continued on next page



q < 0

Figure 11 Figure 12

Table 7: Table summarising general shapes and positions of graphs of functions of the form y = ax+ q.


For f (x) = ax+ q, the domain is {x : x ∈ R} because there is no value of x ∈ R for which f (x) is unde�ned.The range of f (x) = ax + q is also {f (x) : f (x) ∈ R} because there is no value of f (x) ∈ R for which

f (x) is unde�ned.For example, the domain of g (x) = x − 1 is {x : x ∈ R} because there is no value of x ∈ R for which

g (x) is unde�ned. The range of g (x) is {g (x) : g (x) ∈ R}.

5.1.3 Intercepts

For functions of the form, y = ax + q, the details of calculating the intercepts with the x and y axis aregiven.

The y-intercept is calculated as follows:

y = ax+ q

yint = a (0) + q

= q

(12)

For example, the y-intercept of g (x) = x− 1 is given by setting x = 0 to get:

g (x) = x− 1

yint = 0− 1

= −1

(12)

The x-intercepts are calculated as follows:

y = ax+ q

0 = a · xint + q

a · xint = −qxint = − q

a

(12)

For example, the x-intercepts of g (x) = x− 1 is given by setting y = 0 to get:

g (x) = x− 1

0 = xint − 1

xint = 1

(12)



5.1.4 Turning Points

The graphs of straight line functions do not have any turning points.

5.1.5 Axes of Symmetry

The graphs of straight-line functions do not, generally, have any axes of symmetry.

5.1.6 Sketching Graphs of the Form f (x) = ax+ q

In order to sketch graphs of the form, f (x) = ax+ q, we need to determine three characteristics:

1. sign of a2. y-intercept3. x-intercept

Only two points are needed to plot a straight line graph. The easiest points to use are the x-intercept (wherethe line cuts the x-axis) and the y-intercept.

For example, sketch the graph of g (x) = x− 1. Mark the intercepts.Firstly, we determine that a > 0. This means that the graph will have an upward slope.The y-intercept is obtained by setting x = 0 and was calculated earlier to be yint = −1. The x-intercept

is obtained by setting y = 0 and was calculated earlier to be xint = 1.

Figure 13: Graph of the function g (x) = x− 1

Exercise 3: Drawing a straight line graph (Solution on p. 31.)

Draw the graph of y = 2x+ 2

5.1.6.1 Intercepts

1. List the y-intercepts for the following straight-line graphs:

a. y = xb. y = x− 1c. y = 2x− 1d. y + 1 = 2x


2. Give the equation of the illustrated graph below:

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Figure 14


3. Sketch the following relations on the same set of axes, clearly indicating the intercepts with the axesas well as the co-ordinates of the point of interception of the graph: x+2y− 5 = 0 and 3x− y− 1 = 0Click here for the solution14

5.2 Functions of the Form y = ax2 + q

The general shape and position of the graph of the function of the form f (x) = ax2 + q, called a parabola,is shown in Figure 15. These are parabolic functions.

Figure 15: Graph of f (x) = x2 − 1.

5.2.1 Investigation : Functions of the Form y = ax2 + q


a. a (x) = −2 · x2 + 1b. b (x) = −1 · x2 + 1c. c (x) = 0 · x2 + 1d. d (x) = 1 · x2 + 1e. e (x) = 2 · x2 + 1

Use your results to deduce the e�ect of a.2. On the same set of axes, plot the following graphs:

a. f (x) = x2 − 2b. g (x) = x2 − 1c. h (x) = x2 + 0d. j (x) = x2 + 1e. k (x) = x2 + 2

Use your results to deduce the e�ect of q.

The e�ects of a and q are a few examples are what are collectively called transformations of the functiony = x2. The e�ect of q is what is called a translation because all points are moved the same distance inthe same direction, speci�cally a vertical translation (because it slides the graph up and down) of x2 by

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q (note that a horizontal translation by q would look like y = (x− q)2). The e�ect of a is what is called

scaling. When a > 1, it is usually called a stretch because the e�ect on the graph is to stretch it apart,and when a < 1 (but greater than zero), it is often called a shrink because it makes the graph shrink in onitself. When a = −1, it is called a re�ection, speci�cally a re�ection about the x-axis because the x-axis actsas a mirror for the graph. If a is some other negative number, one would refer to it as a scaled re�ection

(or, alternatively, a 'stretched' or 'shrunk' re�ection), because it is re�ected and scaled. One often refers tore�ections, translations, scaling, etc., as the e�ect of the transformation on the given function.

A more in-depth look at transformations can be found here. This section on transformations is simplyan extension and is not needed for examinations.

Complete the following table of values for the functions a to k to help with drawing the required graphsin this activity:

x − 2 − 1 0 1 2

a (x)

b (x)

c (x)

d (x)

e (x)

f (x)

g (x)

h (x)

j (x)

k (x)

Table 8

This simulation allows you to visualise the e�ect of changing a and q. Note that in this simulation q =c. Also an extra term bx has been added in. You can leave bx as 0, or you can also see what e�ect this hason the graph.

Phet simulation for graphing

This media object is a Flash object. Please view or download it at<https://legacy.cnx.org/content/m38379/1.3/equation-grapher.swf>

Figure 16

From your graphs, you should have found that a a�ects whether the graph makes a smile or a frown. Ifa < 0, the graph makes a frown and if a > 0 then the graph makes a smile. This is shown in Figure 17.



Figure 17: Distinctive shape of graphs of a parabola if a > 0 and a < 0.

You should have also found that the value of q a�ects whether the turning point is to the left of they-axis (q > 0) or to the right of the y-axis (q < 0).

These di�erent properties are summarised in Table 9.

a > 0 a < 0

q > 0

Figure 18 Figure 19

q < 0

Figure 20 Figure 21

Table 9: Table summarising general shapes and positions of functions of the form y = ax2 + q.


For f (x) = ax2+q, the domain is {x : x ∈ R} because there is no value of x ∈ R for which f (x) is unde�ned.The range of f (x) = ax2+ q depends on whether the value for a is positive or negative. We will consider

these two cases separately.



If a > 0 then we have:

x2 ≥ 0 (The square of an expression is always positive)

ax2 ≥ 0 (Multiplication by a positive number maintains the nature of the inequality)

ax2 + q ≥ q

f (x) ≥ q

(21)

This tells us that for all values of x, f (x) is always greater than q. Therefore if a > 0, the range off (x) = ax2 + q is {f (x) : f (x) ∈ [q,∞)}.

Similarly, it can be shown that if a < 0 that the range of f (x) = ax2 + q is {f (x) : f (x) ∈ (−∞, q]}.This is left as an exercise.

For example, the domain of g (x) = x2 + 2 is {x : x ∈ R} because there is no value of x ∈ R for whichg (x) is unde�ned. The range of g (x) can be calculated as follows:

x2 ≥ 0

x2 + 2 ≥ 2

g (x) ≥ 2

(21)

Therefore the range is {g (x) : g (x) ∈ [2,∞)}.

5.2.3 Intercepts

For functions of the form, y = ax2+q, the details of calculating the intercepts with the x and y axis is given.The y-intercept is calculated as follows:

y = ax2 + q

yint = a(0)2+ q

= q

(21)

For example, the y-intercept of g (x) = x2 + 2 is given by setting x = 0 to get:

g (x) = x2 + 2

yint = 02 + 2

= 2

(21)

The x-intercepts are calculated as follows:

y = ax2 + q

0 = ax2int + q

ax2int = −q

xint = ±√− q

a

(21)

However, (21) is only valid if − qa ≥ 0 which means that either q ≤ 0 or a < 0. This is consistent with what

we expect, since if q > 0 and a > 0 then − qa is negative and in this case the graph lies above the x-axis and

therefore does not intersect the x-axis. If however, q > 0 and a < 0, then − qa is positive and the graph is

hat shaped and should have two x-intercepts. Similarly, if q < 0 and a > 0 then − qa is also positive, and the

graph should intersect with the x-axis.If q = 0 then we have one intercept at x = 0.



For example, the x-intercepts of g (x) = x2 + 2 is given by setting y = 0 to get:

g (x) = x2 + 2

0 = x2int + 2

−2 = x2int

(21)

which is not real. Therefore, the graph of g (x) = x2 + 2 does not have any x-intercepts.

5.2.4 Turning Points

The turning point of the function of the form f (x) = ax2+q is given by examining the range of the function.We know that if a > 0 then the range of f (x) = ax2 + q is {f (x) : f (x) ∈ [q,∞)} and if a < 0 then therange of f (x) = ax2 + q is {f (x) : f (x) ∈ (−∞, q]}.

So, if a > 0, then the lowest value that f (x) can take on is q. Solving for the value of x at which f (x) = qgives:

q = ax2tp + q

0 = ax2tp

0 = x2tp

xtp = 0

(21)

∴x = 0 at f (x) = q. The co-ordinates of the (minimal) turning point is therefore (0, q).Similarly, if a < 0, then the highest value that f (x) can take on is q and the co-ordinates of the (maximal)

turning point is (0, q).

5.2.5 Axes of Symmetry

There is one axis of symmetry for the function of the form f (x) = ax2 + q that passes through the turningpoint. Since the turning point lies on the y-axis, the axis of symmetry is the y-axis.

5.2.6 Sketching Graphs of the Form f (x) = ax2 + q

In order to sketch graphs of the form, f (x) = ax2 + q, we need to determine �ve characteristics:

1. sign of a2. domain and range3. turning point4. y-intercept5. x-intercept

For example, sketch the graph of g (x) = − 12x

2−3. Mark the intercepts, turning point and axis of symmetry.Firstly, we determine that a < 0. This means that the graph will have a maximal turning point.The domain of the graph is {x : x ∈ R} because f (x) is de�ned for all x ∈ R. The range of the graph is

determined as follows:

x2 ≥ 0

− 12x

2 ≤ 0

− 12x

2 − 3 ≤ −3∴ f (x) ≤ −3

(21)



Therefore the range of the graph is {f (x) : f (x) ∈ (−∞,−3]}.Using the fact that the maximum value that f (x) achieves is -3, then the y-coordinate of the turning

point is -3. The x-coordinate is determined as follows:

− 12x

2 − 3 = −3− 1

2x2 − 3 + 3 = 0

− 12x

2 = 0

Divide both sides by − 12 : x2 = 0

Take square root of both sides :x = 0

∴ x = 0

(21)

The coordinates of the turning point are: (0;−3).The y-intercept is obtained by setting x = 0. This gives:

yint = − 12 (0)

2 − 3

= − 12 (0)− 3

= −3

(21)

The x-intercept is obtained by setting y = 0. This gives:

0 = − 12x

2int − 3

3 = − 12x

2int

−3 · 2 = x2int

−6 = x2int

(21)

which is not real. Therefore, there are no x-intercepts which means that the function does not cross or eventouch the x-axis at any point.

We also know that the axis of symmetry is the y-axis.Finally, we draw the graph. Note that in the diagram only the y-intercept is marked. The graph has

a maximal turning point (i.e. makes a frown) as determined from the sign of a, there are no x-interceptsand the turning point is that same as the y-intercept. The domain is all real numbers and the range is{f (x) : f (x) ∈ (−∞,−3]}.

Figure 22: Graph of the function f (x) = − 12x2 − 3

Exercise 4: Drawing a parabola (Solution on p. 31.)

Draw the graph of y = 3x2 + 5.

The following video shows one method of graphing parabolas. Note that in this video the term vertex isused in place of turning point. The vertex and the turning point are the same thing.



Khan academy video on graphing parabolas - 1

This media object is a Flash object. Please view or download it at<http://www.youtube.com/v/TgKBc3Igx1I&rel=0&hl=en_US&feature=player_embedded&version=3>

Figure 23

5.2.6.1 Parabolas

1. Show that if a < 0 that the range of f (x) = ax2 + q is {f (x) : f (x) ∈ (−∞; q]}. Click here for thesolution15

2. Draw the graph of the function y = −x2 + 4 showing all intercepts with the axes. Click here for thesolution16

3. Two parabolas are drawn: g : y = ax2 + p and h : y = bx2 + q.

Figure 24

a. Find the values of a and p.b. Find the values of b and q.c. Find the values of x for which ax2 + p ≥ bx2 + q.d. For what values of x is g increasing ?


5.3 Functions of the Form y = ax + q

Functions of the form y = ax + q are known as hyperbolic functions. The general form of the graph of this

function is shown in Figure 25.

Figure 25: General shape and position of the graph of a function of the form f (x) = ax+ q.

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5.3.1 Investigation : Functions of the Form y = ax + q


a. a (x) = −2x + 1

b. b (x) = −1x + 1

c. c (x) = 0x + 1

d. d (x) = +1x + 1

e. e (x) = +2x + 1


a. f (x) = 1x − 2

b. g (x) = 1x − 1

c. h (x) = 1x + 0

d. j (x) = 1x + 1

e. k (x) = 1x + 2


You should have found that the value of a a�ects whether the graph is located in the �rst and third quadrantsof Cartesian plane.

You should have also found that the value of q a�ects whether the graph lies above the x-axis (q > 0) orbelow the x-axis (q < 0).

These di�erent properties are summarised in Table 10. The axes of symmetry for each graph are shownas a dashed line.

a > 0 a < 0

q > 0

Figure 26 Figure 27

continued on next page



q < 0

Figure 28 Figure 29

Table 10: Table summarising general shapes and positions of functions of the form y = ax + q. The axes of

symmetry are shown as dashed lines.


For y = ax + q, the function is unde�ned for x = 0. The domain is therefore {x : x ∈ R, x 6= 0}.

We see that y = ax + q can be re-written as:

y = ax + q

y − q = ax

If x 6= 0 then : (y − q)x = a

x = ay−q

(29)

This shows that the function is unde�ned at y = q. Therefore the range of f (x) = ax + q is {f (x) : f (x) ∈

(−∞; q) ∪ (q;∞)}.For example, the domain of g (x) = 2

x + 2 is {x : x ∈ R, x 6= 0} because g (x) is unde�ned at x = 0.

y = 2x + 2

(y − 2) = 2x

If x 6= 0 then : x (y − 2) = 2

x = 2y−2

(29)

We see that g (x) is unde�ned at y = 2. Therefore the range is {g (x) : g (x) ∈ (−∞; 2) ∪ (2;∞)}.

5.3.3 Intercepts

For functions of the form, y = ax + q, the intercepts with the x and y axis is calculated by setting x = 0 for

the y-intercept and by setting y = 0 for the x-intercept.The y-intercept is calculated as follows:

y = ax + q

yint = a0 + q

(29)

which is unde�ned because we are dividing by 0. Therefore there is no y-intercept.For example, the y-intercept of g (x) = 2

x + 2 is given by setting x = 0 to get:

y = 2x + 2

yint = 20 + 2

(29)



which is unde�ned.The x-intercepts are calculated by setting y = 0 as follows:

y = ax + q

0 = axint

+ q

axint

= −qa = −q (xint)

xint = a−q

(29)

For example, the x-intercept of g (x) = 2x + 2 is given by setting x = 0 to get:

y = 2x + 2

0 = 2xint

+ 2

−2 = 2xint

−2 (xint) = 2

xint = 2−2

xint = −1

(29)

5.3.4 Asymptotes

There are two asymptotes for functions of the form y = ax +q. Just a reminder, an asymptote is a straight or

curved line, which the graph of a function will approach, but never touch. They are determined by examiningthe domain and range.

We saw that the function was unde�ned at x = 0 and for y = q. Therefore the asymptotes are x = 0 andy = q.

For example, the domain of g (x) = 2x + 2 is {x : x ∈ R, x 6= 0} because g (x) is unde�ned at x = 0. We

also see that g (x) is unde�ned at y = 2. Therefore the range is {g (x) : g (x) ∈ (−∞; 2) ∪ (2;∞)}.From this we deduce that the asymptotes are at x = 0 and y = 2.

5.3.5 Sketching Graphs of the Form f (x) = ax + q

In order to sketch graphs of functions of the form, f (x) = ax + q, we need to determine four characteristics:

1. domain and range2. asymptotes3. y-intercept4. x-intercept

For example, sketch the graph of g (x) = 2x + 2. Mark the intercepts and asymptotes.

We have determined the domain to be {x : x ∈ R, x 6= 0} and the range to be {g (x) : g (x) ∈ (−∞; 2) ∪(2;∞)}. Therefore the asymptotes are at x = 0 and y = 2.

There is no y-intercept and the x-intercept is xint = −1.



Figure 30: Graph of g (x) = 2x+ 2.

Exercise 5: Drawing a hyperbola (Solution on p. 34.)

Draw the graph of y = −4x + 7.

5.3.5.1 Graphs

1. Using graph (grid) paper, draw the graph of xy = −6.a. Does the point (-2; 3) lie on the graph ? Give a reason for your answer.b. Why is the point (-2; -3) not on the graph ?c. If the x-value of a point on the drawn graph is 0,25, what is the corresponding y-value ?d. What happens to the y-values as the x-values become very large ?e. With the line y = −x as line of symmetry, what is the point symmetrical to (-2; 3) ?


2. Draw the graph of xy = 8.

a. How would the graph y = 83 + 3 compare with that of xy = 8? Explain your answer fully.

b. Draw the graph of y = 83 + 3 on the same set of axes.


5.4 Functions of the Form y = ab(x) + q

Functions of the form y = ab(x) + q are known as exponential functions. The general shape of a graph of afunction of this form is shown in Figure 31.

Figure 31: General shape and position of the graph of a function of the form f (x) = ab(x) + q.

5.4.1 Investigation : Functions of the Form y = ab(x) + q


a. a (x) = −2 · b(x) + 1b. b (x) = −1 · b(x) + 1

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c. c (x) = 0 · b(x) + 1d. d (x) = 1 · b(x) + 1e. e (x) = 2 · b(x) + 1


a. f (x) = 1 · b(x) − 2b. g (x) = 1 · b(x) − 1c. h (x) = 1 · b(x) + 0d. j (x) = 1 · b(x) + 1e. k (x) = 1 · b(x) + 2


You should have found that the value of a a�ects whether the graph curves upwards (a > 0) or curvesdownwards (a < 0).

You should have also found that the value of q a�ects the position of the y-intercept.These di�erent properties are summarised in Table 11.

a > 0 a < 0

q > 0

Figure 32 Figure 33

q < 0

Figure 34 Figure 35

Table 11: Table summarising general shapes and positions of functions of the form y = ab(x) + q.


For y = ab(x) + q, the function is de�ned for all real values of x. Therefore, the domain is {x : x ∈ R}.The range of y = ab(x) + q is dependent on the sign of a.



If a > 0 then:

b(x) ≥ 0

a · b(x) ≥ 0

a · b(x) + q ≥ q

f (x) ≥ q

(35)

Therefore, if a > 0, then the range is {f (x) : f (x) ∈ [q;∞)}.If a < 0 then:

b(x) ≤ 0

a · b(x) ≤ 0

a · b(x) + q ≤ q

f (x) ≤ q

(35)

Therefore, if a < 0, then the range is {f (x) : f (x) ∈ (−∞; q]}.For example, the domain of g (x) = 3 · 2x + 2 is {x : x ∈ R}. For the range,

2x ≥ 0

3 · 2x ≥ 0

3 · 2x + 2 ≥ 2

(35)

Therefore the range is {g (x) : g (x) ∈ [2;∞)}.

5.4.3 Intercepts

For functions of the form, y = ab(x) + q, the intercepts with the x and y axis is calculated by setting x = 0for the y-intercept and by setting y = 0 for the x-intercept.

The y-intercept is calculated as follows:

y = ab(x) + q

yint = ab(0) + q

= a (1) + q

= a+ q

(35)

For example, the y-intercept of g (x) = 3 · 2x + 2 is given by setting x = 0 to get:

y = 3 · 2x + 2

yint = 3 · 20 + 2

= 3 + 2

= 5

(35)



The x-intercepts are calculated by setting y = 0 as follows:

y = ab(x) + q

0 = ab(xint) + q

ab(xint) = −qb(xint) = − q

a

(35)

Which only has a real solution if either a < 0 or q < 0. Otherwise, the graph of the function of formy = ab(x) + q does not have any x-intercepts.

For example, the x-intercept of g (x) = 3 · 2x + 2 is given by setting y = 0 to get:

y = 3 · 2x + 2

0 = 3 · 2xint + 2

−2 = 3 · 2xint

2xint = −23

(35)

which has no real solution. Therefore, the graph of g (x) = 3 · 2x + 2 does not have any x-intercepts.

5.4.4 Asymptotes

Functions of the form y = ab(x) + q have a single horizontal asymptote. The asymptote can be determinedby examining the range.

We have seen that the range is controlled by the value of q. If a > 0, then the range is {f (x) : f (x) ∈[q;∞)}.And if a > 0, then the range is {f (x) : f (x) ∈ [q;∞)}.

This shows that the function tends towards the value of q as x→ −∞. Therefore the horizontal asymptotelies at x = q.

5.4.5 Sketching Graphs of the Form f (x) = ab(x) + q

In order to sketch graphs of functions of the form, f (x) = ab(x)+q, we need to determine four characteristics:

1. domain and range2. asymptote3. y-intercept4. x-intercept

For example, sketch the graph of g (x) = 3 · 2x + 2. Mark the intercepts.We have determined the domain to be {x : x ∈ R} and the range to be {g (x) : g (x) ∈ [2,∞)}.The y-intercept is yint = 5 and there are no x-intercepts.

Figure 36: Graph of g (x) = 3 · 2x + 2.

Exercise 6: Drawing an exponential graph (Solution on p. 34.)

Draw the graph of y = −2.3x + 5.



5.4.5.1 Exponential Functions and Graphs

1. Draw the graphs of y = 2x and y =(12

)xon the same set of axes.

a. Is the x-axis and asymptote or and axis of symmetry to both graphs ? Explain your answer.b. Which graph is represented by the equation y = 2−x ? Explain your answer.c. Solve the equation 2x =

(12

)xgraphically and check that your answer is correct by using substi-

tution.d. Predict how the graph y = 2.2x will compare to y = 2x and then draw the graph of y = 2.2x on

the same set of axes.


2. The curve of the exponential function f in the accompanying diagram cuts the y-axis at the point A(0;1) and B(2; 4) is on f .

Figure 37

a. Determine the equation of the function f .b. Determine the equation of h, the function of which the curve is the re�ection of the curve of f in

the x-axis.c. Determine the range of h.


6 Summary

• You should know the following charecteristics of functions:

· The given or chosen x-value is known as the independent variable, because its value can be chosenfreely. The calculated y-value is known as the dependent variable, because its value depends onthe chosen x-value.

· The domain of a relation is the set of all the x values for which there exists at least one y valueaccording to that relation. The range is the set of all the y values, which can be obtained usingat least one x value.

· The intercept is the point at which a graph intersects an axis. The x-intercepts are the points atwhich the graph cuts the x-axis and the y-intercepts are the points at which the graph cuts they-axis.

· Only for graphs of functions whose highest power is more than 1. There are two types of turningpoints: a minimal turning point and a maximal turning point. A minimal turning point is apoint on the graph where the graph stops decreasing in value and starts increasing in value anda maximal turning point is a point on the graph where the graph stops increasing in value andstarts decreasing.

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· An asymptote is a straight or curved line, which the graph of a function will approach, but nevertouch.

· A line about which the graph is symmetric· The interval on which a graph increases or decreases· A graph is said to be continuous if there are no breaks in the graph.

• Set notation A set of certain x values has the following form: {x : conditions, more conditions}• Interval notation Here we write an interval in the form 'lower bracket, lower number, comma, upper

number, upper bracket'• You should know the following functions and their properties:

· Functions of the form y = ax+ q. These are straight lines.· Functions of the Form y = ax2 + q These are known as parabolic functions or parabolas.· Functions of the Form y = a

x + q. These are known as hyperbolic functions.

· Functions of the Form y = ab(x) + q. These are known as exponential functions.

7 End of Chapter Exercises

1. Sketch the following straight lines:

a. y = 2x+ 4b. y − x = 0c. y = − 1

2x+ 2


2. Sketch the following functions:

a. y = x2 + 3b. y = 1

2x2 + 4

c. y = 2x2 − 4


3. Sketch the following functions and identify the asymptotes:

a. y = 3x + 2b. y = −4.2x + 1c. y = 2.3x − 2


4. Sketch the following functions and identify the asymptotes:

a. y = 3x + 4

b. y = 1x

c. y = 2x − 2


5. Determine whether the following statements are true or false. If the statement is false, give reasonswhy:

a. The given or chosen y-value is known as the independent variable.b. An intercept is the point at which a graph intersects itself.c. There are two types of turning points � minimal and maximal.d. A graph is said to be congruent if there are no breaks in the graph.e. Functions of the form y = ax+ q are straight lines.

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f. Functions of the form y = ax + q are exponential functions.

g. An asymptote is a straight or curved line which a graph will intersect once.h. Given a function of the form y = ax+ q , to �nd the y-intersect put x = 0 and solve for y.i. The graph of a straight line always has a turning point.


6. Given the functions f (x) = −2x2 − 18 and g (x) = −2x+ 6

a. Draw f and g on the same set of axes.b. Calculate the points of intersection of f and g.c. Hence use your graphs and the points of intersection to solve for x when:

i. f (x) > 0

ii. f(x)g(x) ≤ 0

d. Give the equation of the re�ection of f in the x-axis.


7. After a ball is dropped, the rebound height of each bounce decreases. The equation y = 5 ·(0, 8)x showsthe relationship between x, the number of bounces, and y, the height of the bounce, for a certain ball.What is the approximate height of the �fth bounce of this ball to the nearest tenth of a unit ?Click here for the solution28

8. Mark had 15 coins in �ve Rand and two Rand pieces. He had 3 more R2-coins than R5-coins. Hewrote a system of equations to represent this situation, letting x represent the number of �ve randcoins and y represent the number of two rand coins. Then he solved the system by graphing.

a. Write down the system of equations.b. Draw their graphs on the same set of axes.c. What is the solution?


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Solutions to Exercises in this Module

Solution to Exercise (p. 5)

Step 1.

f (n) = n2 − 6n+ 9

f (k − 1) = (k − 1)2 − 6 (k − 1) + 9

(37)

Step 2.

= k2 − 2k + 1− 6k + 6 + 9

= k2 − 8k + 16(37)

We have now simpli�ed the function in terms of k.


Step 1.

f (b) = b2 − 4

u≈ f (b) = 45(37)

Step 2.

b2 − 4 = 45

b2 − 49 = 0

b = +7 or − 7

(37)


Step 1. To �nd the intercept on the y-axis, let x = 0

y = 2 (0) + 2

= 2(37)

Step 2. For the intercept on the x-axis, let y = 0

0 = 2x+ 2

2x = −2x = −1

(37)

Step 3.

Figure 38


Step 1. The sign of a is positive. The parabola will therefore have a minimal turning point.Step 2. The domain is: {x : x ∈ R} and the range is: {f (x) : f (x) ∈ [5,∞)}.Step 3. The turning point occurs at (0, q). For this function q = 5, so the turning point is at (0, 5)



Step 4. The y-intercept occurs when x = 0. Calculating the y-intercept gives:

y = 3x2 + 5

yint = 3 (0)2+ 5

yint = 5

(38)

Step 5. The x-intercepts occur when y = 0. Calculating the x-intercept gives:

y = 3x2 + 5

0 = 3x2 + 5

x2 = − 35

(38)

which is not real, so there are no x-intercepts.Step 6. Putting all this together gives us the following graph:



Figure 39http://cnx.org/content/m38379/1.3/



Step 1. The domain is: {x : x ∈ R, x 6= 0} and the range is: {f (x) : f (x) ∈ (−∞; 7) ∪ (7;∞)}.Step 2. We look at the domain and range to determine where the asymptotes lie. From the domain we see

that the function is unde�ned when x = 0, so there is one asymptote at x = 0. The other asymptoteis found from the range. The function is unde�ned at y = q and so the second asymptote is at y = 7

Step 3. There is no y-intercept for graphs of this form.Step 4. The x-intercept occurs when y = 0. Calculating the x-intercept gives:

y = −4x + 7

0 = −4x + 7

−7 = −4x

xint = 47

(39)

So there is one x-intercept at(47 , 0).

Step 5. Putting all this together gives us the following graph:

Figure 40


Step 1. The domain is: {x : x ∈ R} and the range is: {f (x) : f (x) ∈ (−∞; 5]}.Step 2. There is one asymptote for functions of this form. This occurs at y = q. So the asymptote for this

graph is at y = 5Step 3. The y-intercept occurs when x = 0.

y = −2.3x + 5

y = −2.30 + 5

y = −2 (1) + 5

yint = 7

(40)

So there is one y-intercept at (0, 7).Step 4. The x-intercept occurs when y = 0. Calculating the x-intercept gives:

y = −2.3x + 5

0 = −2.3x + 5

−5 = −2.3x

3xint = 52

xint = 0, 83

(40)

So there is one x-intercept at (0, 83, 0).



Step 5. Putting all this together gives us the following graph:

Figure 41


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Functions and graphs - Grade 10 [CAPS] - CNX